Today, the evidence is strongly in support of the conclusion that complementarity is not nonsense, but the implication is even more shocking. Complementarity is telling us that what happens inside the horizon is as valid a picture of physical reality as what happens outside. The two pictures are equivalent descriptions of the same physics. The inside of a black hole, in other words, is somehow ‘the same’ as the outside. This idea has become known as the holographic principle.
At first sight there might seem to be no need to be so radical as to invoke the holographic principle because we could imagine that, as something falls through the horizon, it is secretly copied. One copy continues to fall into the singularity to be spaghettified and the other copy gets burnt up on the horizon and encoded in the Hawking radiation. Radical as it may be, copying on the horizon does seem less radical than invoking the idea that the interior is a hologram. There is a serious flaw in this logic though – the laws of quantum physics forbid it. The ‘no cloning theorem’ says that it is not possible to make an identical copy of some unknown quantum state, and in Box 13.1 we sketch a proof.
BOX 13.1. No cloning
Suppose we have a cloning machine that can duplicate an unknown qubit. Specifically, this machine takes a |0⟩ and turns it into |0⟩ |0⟩ and likewise it turns a |1⟩ into a |1⟩ |1⟩. What would it do to the following qubit: |Q⟩ = 1/√2(|0⟩ + |1⟩)? This qubit is a 50–50 mix of |0⟩ and |1⟩ and our cloning machine would turn it into 1/√2(|0⟩|0⟩ + |1⟩|1⟩). But this two-qubit state is not |Q⟩ |Q⟩. In other words, our machine is not a cloning machine after all.
With cloning ruled out, it appears that we are left with holography if we want to respect the foundations of both quantum theory and general relativity. There is, however, another possibility: the black hole has no interior. This radical solution would mean general relativity is wrong because nothing can fall into a black hole, which is a gross violation of the Equivalence Principle. This outrage to Einstein was taken very seriously after a 2013 paper by Ahmed Almheiri, Donald Marolf, Joseph Polchinski and James Sully, provocatively titled ‘Black Holes: Complementarity or Firewalls?’41 The AMPS paper, as it has since become known, found what appeared to be a fatal flaw in the complementarity idea. This led to the proposal that a black hole has no inside, and that anyone unfortunate enough to reach the horizon of a black hole would be burnt up in a wall of fire, even from their own point of view.
Firewalls
In Chapter 12, we imagined loitering outside a black hole and collecting the evidence of an astronaut’s fate as they get burnt up on the horizon. We would then jump into the hole to confront the same astronaut with their own ashes. This would generate a contradiction, because the astronaut would have both burnt up and not burnt up from their own perspective. We explained that this contradiction is avoided because the astronaut will have reached the singularity before we are able to catch them. The more precise version of this scenario involves thinking of qubits and cloning. We can imagine throwing a bunch of qubits into the hole and then trying to determine those qubits by collecting the Hawking radiation and processing it so that we have effectively obtained a copy of the original qubits we threw in. To be consistent with the no cloning theorem, it should not be possible to do that and then jump into the hole and meet up with the original qubits. From our understanding of the Page curve, we know that if the black hole is younger than the Page time, not much information will have emerged and we would need to wait a long time (a silly understatement for young, solar mass black holes) before we could obtain a copy of the original qubits. There should be no contradiction, therefore, for a young black hole.
The situation after the Page time, when the black hole is middle-aged or older, is rather more subtle. In that case, the black hole acts more like a mirror and spits the bits back out again almost immediately. That discovery was made in 2007 by Patrick Hayden and John Preskill.42 Surprisingly, however, it turns out that the time delay is still (just) sufficient to prevent a violation of the no cloning theorem. All appears well in the complementarity camp, but AMPS seemed to throw a spanner into the works by coming up with a similar thought experiment. Their scenario, however, cannot be so easily reconciled with complementarity.
Figure 13.1. Illustrating the firewall. For an old black hole, the Hawking particles emitted when the hole was young, R, are entangled with the recently emitted Hawking particle B, which is also entangled with the interior particle A.
In the last chapter, we saw that if information is to be transferred into the Hawking radiation after the Page time, the Hawking particles must gradually become more and more entangled with each other. This is illustrated in the lower half of Figure 12.2. However, the Hawking radiation is produced as entangled pairs, as illustrated in the upper half of Figure 12.2. And here is the problem: because the Hawking pairs are entangled with each other they cannot be entangled with anything else. This is known as the ‘monogamy of entanglement’, and it is another fundamental property of quantum mechanics.
Figure 13.1 illustrates the problem this causes for a black hole older than the Page time. Imagine two observers, Alice and Bob. Bob, sitting outside of the black hole, collects the Hawking radiation. He processes R, the radiation emitted early in the life of the hole (before the Page time), and distils it into a single qubit.† Now, if information is to be conserved, this qubit is highly likely to be entangled with a Hawking particle B that is emitted late on in the life of the hole – this is why the Page curve goes to zero when the hole finally disappears. Bob therefore concludes that B and R are entangled.
Alice is a freely falling observer who crosses the horizon after the Page time. She will confirm that particle B is entangled with the other half of its Hawking pair, labelled A. To avoid violating the monogamy of entanglement, while still permitting the information to come out in the Hawking radiation,‡ we could suppose that Alice does not confirm A and B to be entangled. This might sound innocuous, but it is not. The consequences of simply removing entanglement like this would be very dramatic: it would create a wall of fire. That’s because it costs energy to destroy entanglement in the vacuum – it is tantamount to tearing open empty space. The resulting firewall wouldn’t merely prevent Alice from entering the interior of the black hole, it would effectively destroy space inside the horizon. The interior of the black hole would not exist.
One might wonder whether a complementarity-style argument might still save the day. Maybe Alice could observe that A is entangled with B and Bob could observe that B is entangled with R with no contradiction because they can never meet to confirm their observations. This is not the case though because there is plenty of time for Bob to confirm that he sees entanglement, and then to dive across the horizon to compare notes with Alice, who would have confirmed that she too sees entanglement across the horizon by the very act of crossing it.
With this chain of reasoning, AMPS appeared to have discovered a genuine contradiction which calls into question the existence of the interior of a black hole. This is possibly what provoked Joseph Polchinski to utter the words that open this chapter. The basic problem can be traced to the fact that complementarity appears to be asking for too much entanglement in order to conserve information as the black hole evaporates and to simultaneously preserve the integrity of the quantum vacuum across the horizon. Complementarity requires the black hole and the Hawking radiation to be in an impossible quantum state after the Page time, by demanding they encode more information than the system can physically support.